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Mirrors > Home > ILE Home > Th. List > xlt2add | Unicode version |
Description: Extended real version of lt2add 8207. Note that ltleadd 8208, which has weaker assumptions, is not true for the extended reals (since fails). (Contributed by Mario Carneiro, 23-Aug-2015.) |
Ref | Expression |
---|---|
xlt2add |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xaddcl 9643 | . . . . . . . 8 | |
2 | 1 | 3ad2ant1 1002 | . . . . . . 7 |
3 | 2 | adantr 274 | . . . . . 6 |
4 | simp1l 1005 | . . . . . . . 8 | |
5 | simp2r 1008 | . . . . . . . 8 | |
6 | xaddcl 9643 | . . . . . . . 8 | |
7 | 4, 5, 6 | syl2anc 408 | . . . . . . 7 |
8 | 7 | adantr 274 | . . . . . 6 |
9 | xaddcl 9643 | . . . . . . . 8 | |
10 | 9 | 3ad2ant2 1003 | . . . . . . 7 |
11 | 10 | adantr 274 | . . . . . 6 |
12 | simp3r 1010 | . . . . . . . 8 | |
13 | 12 | adantr 274 | . . . . . . 7 |
14 | simp1r 1006 | . . . . . . . . 9 | |
15 | 14 | adantr 274 | . . . . . . . 8 |
16 | 5 | adantr 274 | . . . . . . . 8 |
17 | simprl 520 | . . . . . . . 8 | |
18 | xltadd2 9660 | . . . . . . . 8 | |
19 | 15, 16, 17, 18 | syl3anc 1216 | . . . . . . 7 |
20 | 13, 19 | mpbid 146 | . . . . . 6 |
21 | simp3l 1009 | . . . . . . . 8 | |
22 | 21 | adantr 274 | . . . . . . 7 |
23 | 4 | adantr 274 | . . . . . . . 8 |
24 | simp2l 1007 | . . . . . . . . 9 | |
25 | 24 | adantr 274 | . . . . . . . 8 |
26 | simprr 521 | . . . . . . . 8 | |
27 | xltadd1 9659 | . . . . . . . 8 | |
28 | 23, 25, 26, 27 | syl3anc 1216 | . . . . . . 7 |
29 | 22, 28 | mpbid 146 | . . . . . 6 |
30 | 3, 8, 11, 20, 29 | xrlttrd 9592 | . . . . 5 |
31 | 30 | anassrs 397 | . . . 4 |
32 | pnfxr 7818 | . . . . . . . . . . . 12 | |
33 | 32 | a1i 9 | . . . . . . . . . . 11 |
34 | pnfge 9575 | . . . . . . . . . . . 12 | |
35 | 24, 34 | syl 14 | . . . . . . . . . . 11 |
36 | 4, 24, 33, 21, 35 | xrltletrd 9594 | . . . . . . . . . 10 |
37 | npnflt 9598 | . . . . . . . . . . 11 | |
38 | 4, 37 | syl 14 | . . . . . . . . . 10 |
39 | 36, 38 | mpbid 146 | . . . . . . . . 9 |
40 | pnfge 9575 | . . . . . . . . . . . 12 | |
41 | 5, 40 | syl 14 | . . . . . . . . . . 11 |
42 | 14, 5, 33, 12, 41 | xrltletrd 9594 | . . . . . . . . . 10 |
43 | npnflt 9598 | . . . . . . . . . . 11 | |
44 | 14, 43 | syl 14 | . . . . . . . . . 10 |
45 | 42, 44 | mpbid 146 | . . . . . . . . 9 |
46 | xaddnepnf 9641 | . . . . . . . . 9 | |
47 | 4, 39, 14, 45, 46 | syl22anc 1217 | . . . . . . . 8 |
48 | npnflt 9598 | . . . . . . . . 9 | |
49 | 2, 48 | syl 14 | . . . . . . . 8 |
50 | 47, 49 | mpbird 166 | . . . . . . 7 |
51 | 50 | adantr 274 | . . . . . 6 |
52 | oveq2 5782 | . . . . . . 7 | |
53 | mnfxr 7822 | . . . . . . . . . . 11 | |
54 | 53 | a1i 9 | . . . . . . . . . 10 |
55 | mnfle 9578 | . . . . . . . . . . 11 | |
56 | 4, 55 | syl 14 | . . . . . . . . . 10 |
57 | 54, 4, 24, 56, 21 | xrlelttrd 9593 | . . . . . . . . 9 |
58 | nmnfgt 9601 | . . . . . . . . . 10 | |
59 | 24, 58 | syl 14 | . . . . . . . . 9 |
60 | 57, 59 | mpbid 146 | . . . . . . . 8 |
61 | xaddpnf1 9629 | . . . . . . . 8 | |
62 | 24, 60, 61 | syl2anc 408 | . . . . . . 7 |
63 | 52, 62 | sylan9eqr 2194 | . . . . . 6 |
64 | 51, 63 | breqtrrd 3956 | . . . . 5 |
65 | 64 | adantlr 468 | . . . 4 |
66 | simpr 109 | . . . . . 6 | |
67 | mnfle 9578 | . . . . . . . . . 10 | |
68 | 14, 67 | syl 14 | . . . . . . . . 9 |
69 | 54, 14, 5, 68, 12 | xrlelttrd 9593 | . . . . . . . 8 |
70 | nmnfgt 9601 | . . . . . . . . 9 | |
71 | 5, 70 | syl 14 | . . . . . . . 8 |
72 | 69, 71 | mpbid 146 | . . . . . . 7 |
73 | 72 | adantr 274 | . . . . . 6 |
74 | 66, 73 | pm2.21ddne 2391 | . . . . 5 |
75 | 74 | adantlr 468 | . . . 4 |
76 | elxr 9563 | . . . . . 6 | |
77 | 5, 76 | sylib 121 | . . . . 5 |
78 | 77 | adantr 274 | . . . 4 |
79 | 31, 65, 75, 78 | mpjao3dan 1285 | . . 3 |
80 | simpr 109 | . . . 4 | |
81 | 39 | adantr 274 | . . . 4 |
82 | 80, 81 | pm2.21ddne 2391 | . . 3 |
83 | oveq1 5781 | . . . . 5 | |
84 | xaddmnf2 9632 | . . . . . 6 | |
85 | 14, 45, 84 | syl2anc 408 | . . . . 5 |
86 | 83, 85 | sylan9eqr 2194 | . . . 4 |
87 | xaddnemnf 9640 | . . . . . . 7 | |
88 | 24, 60, 5, 72, 87 | syl22anc 1217 | . . . . . 6 |
89 | nmnfgt 9601 | . . . . . . 7 | |
90 | 10, 89 | syl 14 | . . . . . 6 |
91 | 88, 90 | mpbird 166 | . . . . 5 |
92 | 91 | adantr 274 | . . . 4 |
93 | 86, 92 | eqbrtrd 3950 | . . 3 |
94 | elxr 9563 | . . . 4 | |
95 | 4, 94 | sylib 121 | . . 3 |
96 | 79, 82, 93, 95 | mpjao3dan 1285 | . 2 |
97 | 96 | 3expia 1183 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3o 961 w3a 962 wceq 1331 wcel 1480 wne 2308 class class class wbr 3929 (class class class)co 5774 cr 7619 cpnf 7797 cmnf 7798 cxr 7799 clt 7800 cle 7801 cxad 9557 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-addass 7722 ax-i2m1 7725 ax-0id 7728 ax-rnegex 7729 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-ltadd 7736 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-po 4218 df-iso 4219 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-xadd 9560 |
This theorem is referenced by: bldisj 12570 |
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